Observer Geometry D; The Algebraic Structure of Observation Backreaction: Cognitive-Physical Coupling in Quantum Field Theory

This paper, within the axiomatic framework of algebraic quantum field theory,introduces the cognitive degrees of freedom of the observer as endogenous variables, establishing an algebraic structure capable of characterizing the backreactionof the observation act on the quantum system. By constructing an extended algebraO=A⊗C,thephysical system and the cognitive system are unified into a composite system, within which the observation backreaction operator Θ is defined. Wepropose four fundamental axioms for Θ—Hermiticity, Poincaré covariance, localityconstraints, and the positive energy condition—ensuring the operator’s compatibility with the core principles of algebraic quantum field theory while endowingthe observer with a dynamical status. Based on these axioms, the fundamentalcommutation relation [Θ,Θ†] = i¯h∆Θ between Θ and its conjugate is derived, andthe Robertson-Schrödinger inequality yields a quantitative bound for the cognitivephysical uncertainty ∆SC ·∆IA ≥ (¯h/2)|⟨∆Θ⟩|. In the context of curved spacetime,the asymptotic behavior of Θ near the Schwarzschild black hole horizon is explored,suggesting a singularity in its radial component of the form Θr ∼ ¯h/(r −rs), and ageneral form for the correction to the effective temperature of Hawking radiation,Teff = TH1+α⟨Θ†Θ⟩/M2 P +··· , is given. The information flow operator J isdefined, and the evolution equation for the reduced density matrix is formally derived, indicating that under the conditions satisfying the axioms, information maybe conserved and the Page curve could naturally emerge. Renormalization groupanalysis shows that Θ should be treated as a composite operator, with its couplingcoefficient gΘ tending to zero in the infrared (emergence of classical observation)and diverging in the ultraviolet (presaging new physics at the Planck scale). As aconcrete realization of the theory, the observation backreaction operator incorporating cognitive degrees of freedom, Θ = d4xK(x) : ϕ2(x) : ⊗C, is constructedin a free scalar field theory, and its commutator is calculated and its physical significance discussed. The qubit model in the appendix further demonstrates thetransition mechanism from the static classical limit to the time-dependent quantum regime. The algebraic framework established in this paper provides a newmathematical language for understanding the quantum measurement problem, theblack hole information problem, and the role of the observer in quantum gravity.